Complex pseudo-partition functions in the Configurationally-Resolved Super-Transition-Array approach for radiative opacity

TL;DR

This paper extends the CRSTA method for calculating hot-plasma radiative opacity by demonstrating the robustness of a recursive relation involving complex pseudo partition functions, even with imaginary components.

Contribution

It shows that the doubly-recursive relation used in STA remains effective and stable when applied to complex pseudo partition functions in CRSTA, despite the presence of imaginary parts.

Findings

The recursive relation is robust with complex functions.

The relation can be expressed as a rotation of previous vectors.

Numerical instabilities are avoided in the complex case.

Abstract

A few years ago, Kurzweil and Hazak developed the Configurationally Resolved Super-Transition-Arrays (CRSTA) method for the computation of hot-plasma radiative opacity. Their approach, based on a temporal integration, is an important refinement of the standard Super-Transition-Arrays (STA) approach, which enables one to recover the underlying structure of the STAs, made of unresolved transition arrays. The CRSTA formalism relies on the use of complex pseudo partition functions, depending on the considered one-electron jump. In this article, we find that, despite the imaginary part, the doubly-recursive relation which was introduced in the original STA method to avoid problems due to alternating-sign terms in partition functions, is still applicable, robust, efficient, and exempt of numerical instabilities. This was rather unexpected, in particular because of the occurrence of trigonometric functions, or Chebyshev polynomials, which can be either positive or negative. We also show that, in the complex case, the recursion relation can be presented in a form where the vector of real and imaginary parts at a given iteration is therefore obtained by a sum of the rotated previous ones.